# Is Non-Euclidean geometry really "Non"?

The definition of a straight line according to google.

I do not understand why I call these geometries "non-Euclidean". In my view, both hyperbolic and elliptical geometry are just a dimensional reference change of the plane, using the same elements described by Euclid. Both are described by curved planes, that is, analyzed three-dimensionaly. A straight line is no longer a straight line. Perhaps there is a lost axiom that has not been introduced to better define what a line is and not to confuse it with a curve. What I want to mean is that, all of these are the same elements but with another perspective. If we can define what a line really is, maybe we can debunk the axioms denying the parallels axiom. If anyone has understood my doubt, please tell me where I am wrong or if there is truth in my words. Thanks.

• In euclidean geometry the parallel postulate holds, in non-euclidean ones it doesn't. I doubt "dimensional reference change" and/or "analyzed three dimensionality" mean anything... The reason non-euclidean geometries are non-euclidean is extremely concrete and simple. Apr 1, 2018 at 4:47
• I suggest you read a modern exposition of euclidean geometry. Axioms do not define what a line is, for example —lines are undefined concepts, and that is in fact most of the point! Apr 1, 2018 at 4:49
• If Lucky's comment is a correct interpretation of your motivation, it's worth noting that the hyperbolic plane cannot in fact be isometrically embedded into $\mathbb{R}^3$; this is due to Hilbert. Apr 1, 2018 at 5:29
• @spaceisdarkgreen, that is simply not true. There are models of euclidean geometry in which lines are not straight lines. A stupid way to get one is to find any bijection of the plane onto itself which does not preserve straight lines, and define a line to be the image of a usual line under that bijection. There are other, more clever models in which the plane is the inside of a disc, for example, and lines are parts of ellipses. Apr 1, 2018 at 5:43
• @nicolas, you should really read up a bit about what euclidean and non-euclidean geometry actually is. Apr 1, 2018 at 5:46